Swansea recovered from an early Arturo Lupoli goal to stun struggling Norwich.
Mark Fotheringham set Lupoli up for a low 12-yard shot, before sweet build-up from Jordi Gomez and Angel Rangel saw Jason Scotland equalise from close in.
Two quick goals after the break sealed Norwich's fate, Darren Pratley firmly heading in Gomez's corner.
Gomez assisted again to set Ferrie Bodde up from six yards, before Mark Fotheringham made the consolation, his shot deflecting in off Angel Rangel.
Norwich City manager Glenn Roeder told BBC Radio Norfolk:
"It's just diabolically poor play, poor defending, the player who was told to mark the goal scorer was marking on the wrong side, it's totally unacceptable.
"It's so disappointing when they don't do what they're told, It bothers me that some of them are quite experienced players.
"They will perhaps continue to make those errors wherever they play, It's just a total lack of football intelligence," he said.
Swansea manager Roberto Martinez:
"In football everyone must have a dream and ours is to play in the Premier League.
"But it could take us five years or even 10 years to achieve.
"Expectation levels are always very high in south Wales, but at the moment all we are trying to do is to consolidate at this level.
"Obviously I am very happy with the number of points we have won so far but our aim this season is to stay in this division.
"There are six or seven players here who have the ability to play in the Premier League, but at the moment we have still got to prove that we can compete week in, week out at this level.
"It was a big thing for us to come to a place like Norwich and get a result like this.
"This is the first time we have come to a Premier League environment and been ourselves, played the way we can and picked up the three points we deserved. It's a big step forward for us, a big moment."
Norwich: Marshall, Otsemobor, Kennedy, Doherty, Bertrand, Bell, Clingan, Fotheringham (Hoolahan 62), Pattison, Lupoli (Cureton 66), Lita.
Subs Not Used: Nelson, Drury, Omozusi.
Goals: Lupoli 28, Rangel 61 og.
Swansea: Konstantopoulos, Rangel, Williams, Monk, Bessone, Gomez, Britton, Bodde, Pratley, Gower (Pintado 78), Scotland (Brandy 90).
Subs Not Used: Krysiak, Tate, Bauza.
Booked: Bodde.
Goals: Scotland 43, Pratley 47, Bodde 48.
Att: 24,262
Ref: Grant Hegley (Hertfordshire).
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